Let (V,+V,·V) and (W,+W,·W) be vector spaces and define V ⊕ W = {(v,w) : v...
Let (V,+V,·V) and (W,+W,·W) be vector spaces and define
V ⊕ W = {(v,w) : v ∈ V and w ∈ W}. Prove that
(a) V ⊕ W is a vector space, under componentwise operations.
(b) ifS={(v,0):v∈V}andS′ ={(0,w):w∈ W}, then S and S′ are subspaces of V ⊕ W.
Homework Answers
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Let (V,+V,·V) and (W,+W,·W) be vector spaces and define
V ⊕ W = {(v,w) : v...
2. Let V and W be vector spaces over F. Define the set v, w) |v...
2. Let V and W be vector spaces over F. Define the set v, w) |v V andwEW This is called the product of V and W (a) Show that V x W is a vector space. (b) Define a map w : V → V × W by w (z) = (z,0). Show that w is an injective linear map. Note that we can define a similar map lw (c) If (d) Show that V x W. (V W...
Let V be a finite-dimensional vector space over F. For every subset SCV, define Sº =...
Let V be a finite-dimensional vector space over F. For every subset SCV, define Sº = {f EV* | f(s) = 0 Vs E S}. (a) Prove that sº is a subspace of V* (S may not be a subspace!) (b) If W is a subspace of V and x € W, prove that there exists an fe Wº with f(x) + 0. (c) If v inV, define û :V* + F by ū(f) = f(u). (This is linear and...
Let V and W be vector spaces over F, and let f: V W be a...
Let V and W be vector spaces over F, and let f: V W be a linear transformation. (a) Prove that f is one-to-one if and only if f carries linearly independent (b) Suppose that f is one-to-one and that S is a subset of V. Prove that subsets of V to linearlv independent subsets of W S is linearly independent if and only if (S) is linearly independent.
Please answer me fully with the details. Thanks! Let V, W and X be vector spaces....
Please answer me fully with the details. Thanks!
Let V, W and X be vector spaces. Let T: V -> W and S : W -> X be isomorphisms. Prove that SoT : V -+ X is an isomorphism.
Let V, W and X be vector spaces. Let T: V -> W and S : W -> X be isomorphisms. Prove that SoT : V -+ X is an isomorphism.
just part c,d, and e please!! Let V be a finite-dimensional vector space over F. For...
just part c,d, and e please!!
Let V be a finite-dimensional vector space over F. For every subset SCV, define Sº = {f eV" f(s) = 0 Vs ES}. (a) Prove that sº is a subspace of V* (S may not be a subspace!) (b) If W is a subspace of V and r & W, prove that there exists an few with f(x) +0. (c) If v inV, define u:V* → F by 0(f) = f(v). (This is linear...
solution to 2 (ii) Show that the image of f is not a subspace of R 2. Let U, V, and W be vector spaces over the field k, and let f: Ux V- W be a bilinear map. Show that the image of f is a union o...
solution to 2
(ii) Show that the image of f is not a subspace of R 2. Let U, V, and W be vector spaces over the field k, and let f: Ux V- W be a bilinear map. Show that the image of f is a union of subspaces of W. 3. Let k be a field, and let U, V, and W be vector spaces over k. Recall that
(ii) Show that the image of f is not...
Problem 3. Let V and W be vector spaces, let T : V -> W be...
Problem 3. Let V and W be vector spaces, let T : V -> W be a linear transformation, and suppose U is a subspace of W (a) Recall that the inverse image of U under T is the set T-1 U] := {VE V : T(v) E U). Prove that T-[U] is a subspace of V (b) Show that U nim(T) is a subspace of W, and then without using the Rank-Nullity Theorem, prove that dim(T-1[U]) = dim(Unin (T))...
a. Let W and X both be subspaces of a vector space V. Prove that dim(WnX)...
a. Let W and X both be subspaces of a vector space V. Prove that dim(WnX) > dim(W) + dim(X) - dim(V) b. Define a plane in R" (as a vector space) to be any subspace of dimension 2, and a line to be any subspace of dimension 1. Show that the intersection of any two planes in R' contains a line. c. Must the intersection of two planes in R* contain a line?
Let V be a vector space over a field F, and let U and W be...
Let V be a vector space over a field F, and let U and W be finite dimensional subspaces of V. Consider the four subspaces X1 = U, X2 = W, X3 = U+W, X4 = UnW. Determine if dim X; <dim X, or dim X, dim X, or neither, must hold for every choice of i, j = 1,2,3,4. Prove your answers.
(4) Let V and W be vector spaces over R: consider the free vector space F(V × W) on the Cartesian...
(4) Let V and W be vector spaces over R: consider the free vector space F(V × W) on the Cartesian product V x W of V and W. Given an element (v, w) of V x W, we view (v, w) as an element of F(V x W) via the inclusion map i : V x W F(V x W) Any element of F(V x W) is a finite linear combination of such elements (v, w) Warning. F(V ×...
2. Let V and W be vector spaces over F. Define the set v, w) |v V andwEW This is called the product of V and W (a) Show that V x W is a vector space. (b) Define a map w : V → V × W by w (z) = (z,0). Show that w is an injective linear map. Note that we can define a similar map lw (c) If (d) Show that V x W. (V W...
Let V be a finite-dimensional vector space over F. For every subset SCV, define Sº = {f EV* | f(s) = 0 Vs E S}. (a) Prove that sº is a subspace of V* (S may not be a subspace!) (b) If W is a subspace of V and x € W, prove that there exists an fe Wº with f(x) + 0. (c) If v inV, define û :V* + F by ū(f) = f(u). (This is linear and...
Let V and W be vector spaces over F, and let f: V W be a linear transformation. (a) Prove that f is one-to-one if and only if f carries linearly independent (b) Suppose that f is one-to-one and that S is a subset of V. Prove that subsets of V to linearlv independent subsets of W S is linearly independent if and only if (S) is linearly independent.
Please answer me fully with the details. Thanks!
Let V, W and X be vector spaces. Let T: V -> W and S : W -> X be isomorphisms. Prove that SoT : V -+ X is an isomorphism.
Let V, W and X be vector spaces. Let T: V -> W and S : W -> X be isomorphisms. Prove that SoT : V -+ X is an isomorphism.
just part c,d, and e please!!
Let V be a finite-dimensional vector space over F. For every subset SCV, define Sº = {f eV" f(s) = 0 Vs ES}. (a) Prove that sº is a subspace of V* (S may not be a subspace!) (b) If W is a subspace of V and r & W, prove that there exists an few with f(x) +0. (c) If v inV, define u:V* → F by 0(f) = f(v). (This is linear...
solution to 2
(ii) Show that the image of f is not a subspace of R 2. Let U, V, and W be vector spaces over the field k, and let f: Ux V- W be a bilinear map. Show that the image of f is a union of subspaces of W. 3. Let k be a field, and let U, V, and W be vector spaces over k. Recall that
(ii) Show that the image of f is not...
Problem 3. Let V and W be vector spaces, let T : V -> W be a linear transformation, and suppose U is a subspace of W (a) Recall that the inverse image of U under T is the set T-1 U] := {VE V : T(v) E U). Prove that T-[U] is a subspace of V (b) Show that U nim(T) is a subspace of W, and then without using the Rank-Nullity Theorem, prove that dim(T-1[U]) = dim(Unin (T))...
a. Let W and X both be subspaces of a vector space V. Prove that dim(WnX) > dim(W) + dim(X) - dim(V) b. Define a plane in R" (as a vector space) to be any subspace of dimension 2, and a line to be any subspace of dimension 1. Show that the intersection of any two planes in R' contains a line. c. Must the intersection of two planes in R* contain a line?
Let V be a vector space over a field F, and let U and W be finite dimensional subspaces of V. Consider the four subspaces X1 = U, X2 = W, X3 = U+W, X4 = UnW. Determine if dim X; <dim X, or dim X, dim X, or neither, must hold for every choice of i, j = 1,2,3,4. Prove your answers.
(4) Let V and W be vector spaces over R: consider the free vector space F(V × W) on the Cartesian product V x W of V and W. Given an element (v, w) of V x W, we view (v, w) as an element of F(V x W) via the inclusion map i : V x W F(V x W) Any element of F(V x W) is a finite linear combination of such elements (v, w) Warning. F(V ×...
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